Boundary Half-Strips and the Strong CHIP

When the subdifferential sum rule formula holds for the indicator functions ιC and ιD of two closed convex sets C and D of a locally convex space X, the pair (C,D) is said to have the strong conical hull intersection property (the strong CHIP). The specification of a well-known theorem due to Moreau to the case of the support functionals σC and σD subsumes the fact that the pair (C,D) has the strong CHIP whenever the inf-convolution of σC and σD is exact. In this article we prove, in the setting of Euclidean spaces, that if the pair (C,D) has the strong CHIP while the boundary of C does not contain any half-strip, then the inf-convolution of σC and σD is exact. Moreover, when the boundary of a closed and convex set C does contain a half-strip, it is possible to find a closed and convex set D such that the pair (C,D) has the strong CHIP while the inf-convolution of σC and σD is not exact. The validity of the converse of Moreau’s theorem in Euclidean spaces is thus associated to the absence of half-strips within the boundary of concerned convex sets.